Do we have a nice classification of injective modules over a local ring?

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Let $R$ be a local ring. We know, for example, that $V$ is a projective $R$ module if and only if $V$ is free. Also, since $R$ is not semisimple, there must be at least one module which is not projective and one which is not injective.



Is there a way to classify injective modules over a local ring? The best candidate would be the stament "$V$ is injective if and only if it is divisible." However, for example, $Bbb Z/(3)$ is divisible as a $Bbb Z/(9)$ module, but not injective since $Bbb Z/(9)notcong Bbb Z/(3)oplus Bbb Z/(3)$.



My only reason for believing there is such a statement is because the notions are categorically dual. Still, that doesn't imply that there must be some analog, but I thought I'd ask.




soft-question modules local-rings




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    Let $R$ be a local ring. We know, for example, that $V$ is a projective $R$ module if and only if $V$ is free. Also, since $R$ is not semisimple, there must be at least one module which is not projective and one which is not injective.



    Is there a way to classify injective modules over a local ring? The best candidate would be the stament "$V$ is injective if and only if it is divisible." However, for example, $Bbb Z/(3)$ is divisible as a $Bbb Z/(9)$ module, but not injective since $Bbb Z/(9)notcong Bbb Z/(3)oplus Bbb Z/(3)$.



    My only reason for believing there is such a statement is because the notions are categorically dual. Still, that doesn't imply that there must be some analog, but I thought I'd ask.




    soft-question modules local-rings




    share|cite|improve this question





















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      Let $R$ be a local ring. We know, for example, that $V$ is a projective $R$ module if and only if $V$ is free. Also, since $R$ is not semisimple, there must be at least one module which is not projective and one which is not injective.



      Is there a way to classify injective modules over a local ring? The best candidate would be the stament "$V$ is injective if and only if it is divisible." However, for example, $Bbb Z/(3)$ is divisible as a $Bbb Z/(9)$ module, but not injective since $Bbb Z/(9)notcong Bbb Z/(3)oplus Bbb Z/(3)$.



      My only reason for believing there is such a statement is because the notions are categorically dual. Still, that doesn't imply that there must be some analog, but I thought I'd ask.




      soft-question modules local-rings




      share|cite|improve this question











      Let $R$ be a local ring. We know, for example, that $V$ is a projective $R$ module if and only if $V$ is free. Also, since $R$ is not semisimple, there must be at least one module which is not projective and one which is not injective.



      Is there a way to classify injective modules over a local ring? The best candidate would be the stament "$V$ is injective if and only if it is divisible." However, for example, $Bbb Z/(3)$ is divisible as a $Bbb Z/(9)$ module, but not injective since $Bbb Z/(9)notcong Bbb Z/(3)oplus Bbb Z/(3)$.



      My only reason for believing there is such a statement is because the notions are categorically dual. Still, that doesn't imply that there must be some analog, but I thought I'd ask.





      soft-question modules local-rings





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      asked Jul 26 at 19:31









      Elliot G

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